Abstract:

This article mainly investigates the initial boundary value problem of a chemotaxis-convection model pertaining to the growth of capillary sprouts during tumor angiogenesis $$\begin{equation*} \begin{cases} \frac{\partial u}{\partial t}=\Delta u-\chi_1\nabla\cdot(u\nabla v)+\chi_2\nabla\cdot(u\nabla w), &(x,t)\in\Omega\times(0, \infty),\\ \frac{\partial v}{\partial t}=\Delta v+\xi_1\nabla\cdot(v\nabla w) - v + u, &(x,t)\in\Omega\times(0, \infty),\\ \frac{\partial w}{\partial t}=\Delta w - w + u, &(x,t)\in\Omega\times(0, \infty), \end{cases} \end{equation*}$$ where $\Omega\subset\mathbb{R}^N(N\ge1)$ is a smoothly bounded domain with Neumman boundary condition. The parameters $\chi_1$, $\chi_2$ and $\xi_1>0$. We demonstrate that this model admits a globally bounded classical solution under the smallness condition on $\chi_1$ and $\chi_2$. Moreover, this solution will converge exponentially to its constant steady state $(\bar{u}_0,\bar{u}_0,\bar{u}_0)$ under the further smallness condition on $\xi_1$ and $\chi_2$, where $\bar{u}_0=\frac{1}{|\Omega|}\int_\Omega u \mathrm{d}x$.

Key words: tumor angiogenesis, chemotaxis-convection, global boundedness, large time behavior